The model equations follow the principles of mass transport, fluid dynamics, and biochemistry in order to simulate the fate of a substance in the body. Compartments are usually defined by grouping organs or tissues with similar blood perfusion rate and lipid content (
i.e. organs for which chemicals' concentration
vs. time profiles will be similar). Ports of entry (lung, skin, intestinal tract...), ports of exit (kidney, liver...) and target organs for
therapeutic effect or toxicity are often left separate. Bone can be excluded from the model if the substance of interest does not distribute to it. Connections between compartment follow physiology (
e.g., blood flow in exit of the gut goes to liver,
etc.)
Basic transport equations Drug distribution into a tissue can be rate-limited by either perfusion or permeability.
Perfusion-rate-limited kinetics apply when the tissue membranes present no barrier to diffusion. Blood flow, assuming that the drug is transported mainly by blood, as is often the case, is then the limiting factor to distribution in the various cells of the body. That is usually true for small
lipophilic drugs. Under perfusion limitation, the instantaneous rate of entry for the quantity of drug in a compartment is simply equal to (blood)
volumetric flow rate through the organ times the incoming blood concentration. In that case; for a generic compartment
i, the differential equation for the quantity
Qi of substance, which defines the rate of change in this quantity, is: {dQ_i \over dt} = F_i (C_{art} - {{Q_i} \over {P_i V_i}}) where
Fi is blood flow (noted
Q in the Figure above),
Cart incoming
arterial blood concentration,
Pi the tissue over blood
partition coefficient and
Vi the volume of compartment
i. A complete set of differential equations for the 7-compartment model shown above could therefore be given by the following table: The above equations include only transport terms and do not account for inputs or outputs. Those can be modeled with specific terms, as in the following.
Modeling inputs Modeling inputs is necessary to come up with a meaningful description of a chemical's pharmacokinetics. The following examples show how to write the corresponding equations.
Ingestion When dealing with an oral
bolus dose (
e.g. ingestion of a tablet), first order absorption is a very common assumption. In that case the gut equation is augmented with an input term, with an absorption rate constant
Ka: {dQ_g \over dt} = F_g (C_{art} - {{Q_g} \over {P_g V_g}}) + K_a Q_{ing} That requires defining an equation for the quantity ingested and present in the gut lumen: {dQ_{ing} \over dt} = - K_a Q_{ing} In the absence of a gut compartment, input can be made directly in the liver. However, in that case local metabolism in the gut may not be correctly described. The case of approximately continuous absorption (
e.g. via drinking water) can be modeled by a zero-order absorption rate (here
Ring in units of mass over time): {dQ_g \over dt} = F_g (C_{art} - {{Q_g} \over {P_g V_g}}) + R_{ing} More sophisticated gut absorption model can be used. In those models, additional compartments describe the various sections of the gut lumen and tissue. Intestinal pH, transit times and presence of active transporters can be taken into account .
Skin depot The absorption of a chemical deposited on skin can also be modeled using first order terms. It is best in that case to separate the skin from the other tissues, to further differentiate exposed skin and non-exposed skin, and differentiate viable skin (
dermis and epidermis) from the
stratum corneum (the actual skin upper layer exposed). This is the approach taken in [Bois F., Diaz Ochoa J.G. Gajewska M., Kovarich S., Mauch K., Paini A., Péry A., Sala Benito J.V., Teng S., Worth A., in press, Multiscale modelling approaches for assessing cosmetic ingredients safety, Toxicology. doi: 10.1016/j.tox.2016.05.026] Unexposed
stratum corneum simply exchanges with the underlying viable skin by diffusion: {dQ_{{sc}_{u}} \over dt} = K_p \times S_s \times (1 - f_{S_{e}}) \times ({Q_{s_u} \over {P_{sc} V_{{sc}_{u}}}} - C_{{sc}_u}) where K_p is the partition coefficient, S_s is the total skin surface area, f_{S_{e}} the fraction of skin surface area exposed, ... For the viable skin unexposed: {dQ_{s_u} \over dt} = F_s (1 - f_{S_{e}}) (C_{art} - {{Q_{s_u}} \over {P_s V_{s_u}}}) - {dQ_{{sc}_{u}} \over dt} For the skin
stratum corneum exposed: {dQ_{{sc}_{e}} \over dt} = K_p \times S_s \times f_{S_{e}} \times ({Q_{s_e} \over {P_{sc} V_{{sc}_e}}} - C_{{sc}_e}) for the viable skin exposed: {dQ_{s_e} \over dt} = F_s f_{S_{e}} (C_{art} - {{Q_{s_e}} \over {P_s V_{s_e}}}) - {dQ_{{sc}_{e}} \over dt} dt(QSkin_u) and dt(QSkin_e) feed from arterial blood and back to venous blood. More complex diffusion models have been published [reference to add].
Intra-venous injection Intravenous injection is a common clinical
route of administration. (to be completed)
Inhalation Inhalation occurs through the lung and is hardly dissociable from exhalation (to be completed)
Modelling metabolism There are several ways metabolism can be modeled. For some models, a linear excretion rate is preferred. This can be accomplished with a simple differential equation. Otherwise a
Michaelis-Menten equation, as follows, is generally appropriate for a more accurate result. : v = \frac{d [P]}{d t} = \frac{ V_\max {[S]}}{K_m + [S]} . ==Uses of PBPK modeling==